In a sequence of terms, the first term is (-1). Each term thereafter is obtained by multiplying the previous number with (-2). How many of the first 50 terms of the series are less than 50?

Options

A. 3
B. 25
C. 28
D. 32

Answer

To solve the problem, we need to analyze the sequence defined by the first term and the rule for generating subsequent terms. Let's break it down step by step.

### Step 1: Identify the sequence
The first term of the sequence is given as:
\[ a_1 = -1 \]
Each subsequent term is obtained by multiplying the previous term by \(-2\). Thus, we can express the \(n\)-th term of the sequence as:
\[ a_n = (-1) \times (-2)^{n-1} \]

### Step 2: Calculate the first few terms
Let's compute the first few terms of the sequence:
- \( a_1 = -1 \)
- \( a_2 = -1 \times (-2) = 2 \)
- \( a_3 = 2 \times (-2) = -4 \)
- \( a_4 = -4 \times (-2) = 8 \)
- \( a_5 = 8 \times (-2) = -16 \)
- \( a_6 = -16 \times (-2) = 32 \)
- \( a_7 = 32 \times (-2) = -64 \)
- \( a_8 = -64 \times (-2) = 128 \)

The sequence alternates between negative and positive values.

### Step 3: Determine the sign of the terms
From our calculations, we observe:
- Odd-indexed terms (1st, 3rd, 5th, etc.) are negative.
- Even-indexed terms (2nd, 4th, 6th, etc.) are positive.

### Step 4: Count the odd-indexed terms
Since the first 50 terms consist of 25 odd-indexed terms, all of which are negative, we can conclude:
- All 25 odd-indexed terms are less than 50.

### Step 5: Count the even-indexed terms
Now, we need to check the even-indexed terms to see how many of them are less than 50:
- \( a_2 = 2 \) (less than 50)
- \( a_4 = 8 \) (less than 50)
- \( a_6 = 32 \) (less than 50)
- \( a_8 = 128 \) (not less than 50)

Thus, the even-indexed terms that are less than 50 are \(2\), \(8\), and \(32\). There are 3 such terms.

### Step 6: Calculate the total number of terms less than 50
Now we can sum the counts of odd and even terms:
- Odd terms: 25
- Even terms: 3

Total terms less than 50:
\[ 25 + 3 = 28 \]

### Conclusion
The total number of the first 50 terms of the series that are less than 50 is:
\[ \boxed{28} \]